On the distribution of Rudin-Shapiro polynomials and lacunary walks on $SU(2)$

TL;DR

This paper characterizes the limiting distribution of Rudin-Shapiro polynomials, showing they become uniformly distributed in the disc, and introduces a non-commutative central limit theorem related to $SU(2)$.

Contribution

It provides a new understanding of the distribution of Rudin-Shapiro polynomials and develops a non-commutative CLT using $SU(2)$ representation theory.

Findings

Rudin-Shapiro polynomials' values become uniformly distributed in the disc

Established a non-commutative analogue of the classical CLT

Resolved conjectures of Saffari and Montgomery

Abstract

We characterize the limiting distribution of Rudin-Shapiro polynomials, showing that, normalized, their values become uniformly distributed in the disc. This resolves conjectures of Saffari and Montgomery. Our proof proceeds by relating the polynomials' distribution to that of a product of weakly dependent random matrices, which we analyze using the representation theory of $SU(2)$. Our approach leads us to a non-commutative analogue of the classical central limit theorem of Salem and Zygmund, which may be of independent interest.